Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response

Abstract

The fluctuation-dissipation theorem is a cornerstone result in statistical mechanics that can be used to translate the statistics of the free natural variability of a system into information on its forced response to perturbations. By combining this viewpoint on response theory with the key ingredients of Koopmanism, it is possible to deconstruct virtually any response operator into a sum of terms, each associated with a specific mode of natural variability of the system. This dramatically improves the interpretability of the resulting response formulas. We show here on three simple yet mathematically meaningful examples how to use the Extended Dynamical Mode Decomposition (EDMD) algorithm on an individual trajectory of the system to compute with high accuracy correlation functions as well as Green functions associated with acting forcings. This demonstrates the great potential of using Koopman analysis for the key problem of evaluating and testing the sensitivity of a complex system.

References in corpus (34)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Fluctuation-Dissipation: Response Theory in Statistical Physics
• Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
• Applied Koopmanism
• Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• The Physics of Climate Variability and Climate Change
• A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics
• A review of linear response theory for general differentiable dynamical systems
• On the numerical approximation of the Perron-Frobenius and Koopman operator
• An update on nonequilibrium linear response
• A simple framework to justify linear response theory
• Nonparametric forecasting of low-dimensional dynamical systems
• A Statistical Mechanical Approach for the Computation of the Climatic Response to General Forcings
• Predicting Climate Change using Response Theory: Global Averages and Spatial Patterns
• Residual Dynamic Mode Decomposition: Robust and verified Koopmanism
• Reduced-Order Models for Coupled Dynamical Systems: Data-driven Methods and the Koopman Operator
• Beyond the linear Fluctuation-Dissipation Theorem: the Role of Causality
• Crisis of the Chaotic Attractor of a Climate Model: A Transfer Operator Approach
• On the Fluctuation-Dissipation Relation in non-equilibrium and non-Hamiltonian systems
• Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond
• Response Operators for Markov Processes in a Finite State Space: Radius of Convergence and Link to the Response Theory for Axiom A Systems
• On Some Aspects of the Response to Stochastic and Deterministic Forcings
• Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory
• Response and Sensitivity Using Markov Chains
• Complete spectral data for analytic Anosov maps of the torus
• Dynamic mode decomposition for analytic maps
• Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems
• Response Formulae for $n$-point Correlations in Statistical Mechanical Systems and Application to a Problem of Coarse Graining
• Dimension reduction of noisy interacting systems
• Response Theory Identifies Reaction Coordinates and Explains Critical Phenomena in Noisy Interacting Systems
• Explicit examples of resonances for Anosov maps of the torus
• A General Framework for Linking Free and Forced Fluctuations via Koopmanism
• Kolmogorov Modes and Linear Response of Jump-Diffusion Models